Applied Mathematical Sciences, Vol. 5, 2011, no. 12, 595 - 606
A New Algorithm for Optimal Control of Time-Delay Systems H. R. Sharif 1, M. A. Vali2, M. Samavat1 and A. A. Gharavisi1 1
Department of Electrical Engineering Shahid Bahonar University Kerman, Iran
[email protected] (H. R. Sharif)
[email protected] (M. Samavat) gharavisi@ mail.uk.ac.ir (A. A. Gharavisi) 2
Department of Mathematics Shahid Bahonar University Kerman, Iran
[email protected] Abstract
A new method for solving the optimal solution of linear time invariant delay systems using the Legendre wavelets is presented. Using the operational matrix of integration together with the delay matrix Td , the dynamical equation of delay systems is reduced to a set of simultaneous linear algebraic equations. Owing to the simplicity of delay matrix Td (having many zero entries), the method presents considerable computational advantages when compared with the other possible methods. Keywords: Optimal control; Delay systems; Legendre wavelets; Operational matrix
1. Introduction Systems with time delay occur frequently in mechanical and electrical systems, industrial process, population, economic growth, neural networks, etc. Many researchers have tried various methods of optimizing linear time delay systems.
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H. R. Sharif, M. A. Vali, M. Samavat and A. A. Gharavisi
Typical researchers are Kharatishvili [12], Inoue [9], Soliman [27], Jamshidi [10], Malek-Zavarei [13] and Hwang [8]. The orthogonal functions and polynomial series have been developed for solving various problems of dynamical systems such as system analysis, parameter identification, optimal control, etc [18, 1, 5]. The main idea of this technique is that it reduces these problems to those of solving a system of algebraic equations and thus it greatly simplifies the problem. The approach is based on converting the differential equations into an integral equation through integration. The state and/or control involved in the equation are approximated by finite terms of orthogonal series and by using the operational matrix of integration the integral operations are eliminated. The form of the operational matrix of integration depends on the particular choice of the orthogonal functions like Walsh functions [3], Block-pulse functions [7], Laguerre series [25], Jacobi series [19], Fourier series [28], Bessel series [17], Taylor series [20], Shifted Legendre [21], Chebyshev polynomials [16] and Hermite polynomials [26]. In this study, we use wavelet functions to approximate both the control and state functions. It avoids the difficult integral equations created from variational methods reducing the problem to the solution of an algebraic system of equations, thus providing a computationally more efficient approach. wavelet functions possess useful properties such as orthogonality, compact support and exact representation of polynomials to a certain degree such as Haar [11], Legendre [22] and Sine-cosine [23] wavelets. In this paper, Legendre wavelets (LeW) are employed to solve the optimal control of time-delay systems. The operational matrix of integration and time delay matrix are given. These matrices are then used to evaluate the coefficients of the Legendre wavelets in such a way that the necessary conditions for extremity of performance index is imposed. Illustrative examples are given to demonstrate the applicability of the proposed method.
2. Preleminaries and Problem Statement 2.1. Properties of Legendre Wavelets Wavelets constitute a family of functions constructed from dilation and translation of a single function called mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets as [6] 1 t −b ψ a ,b (t ) = ψ( ) a, b ∈ R, a ≠ 0 a a If we restrict the parameters a and b to discrete values as a = a0− k , b = b0 na0− k , a0 > 1, b0 > 0 and n and k positive integers, we have the following family of discrete wavelets: k
Ψk , n (t ) = a0 2 Ψ (a0k t − nb0 )
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where Ψk , n (t ) forms a wavelets basis for L2(R). In particular, when a0 = 2, b0 = 1, then Ψk , n (t ) forms an orthogonal basis [29]. Legendre wavelets Ψn ,m (t ) = Ψ (n, k , m, t ) have four arguments; nˆ = 2n − 1, n = 1,...,2 k −1 , 2 k −1 = 2,3,4,... , m is the order for Legendre polynomials and t is the
normalized time. k ⎧ 1 2 k ⎪ m + 2 p m ( 2 t − nˆ ) 2 ⎪ Ψ n ,m (t ) = ⎨ ⎪ 0 ⎪ ⎩
nˆ − 1 nˆ + 1 ≤t < k k 2 2
(1)
Otherwise
Where m = 0,1,2,... , n = 1,...,2 k −1 . Here, pm (t ) are the well-known Legendre polynomials of order m on the interval [-1,1], and satisfy the following recursive formula [2]: p m +1 (t ) = (
2m + 1 m )tp m (t ) − ( ) p m −1 (t ) m +1 m +1
p0 (t ) = 1, p1 (t ) = t
2.2. Function Approximation A function f (t ) ∈ L2 [0, T ] defined over [0, T ) may be expanded as 2k −1 ∞
f (t ) = ∑ ∑ cn ,mψ n ,m (t )
(2)
n =1 m =0
Where cn ,m = f (t ),ψ n ,m (t ) in which 〈.〉 denotes the inner product of L2 [0, T ] . If the infinite series in Eq.(2) is truncated to m = 0,1,..., M − 1 , then Eq.(2) can be written as 2 k −1 M −1
f (t ) ≈ ∑ ∑ cn ,mψ n,m (t ) = C T Ψ (t )
(3)
n =1 m =0
Where C and Ψ (t ) are 2k −1 M × 1 matrices given by
C = [c1,0 , c1,1 ,..., c1,M −1 , c2,0 ,..., c2,M −1 ,..., c2k −1 ,0 ,..., c2k −1 ,M −1 ]T Ψ(t ) = [ψ1,0 (t ),ψ1,1 (t ),...,ψ1,M −1 (t ),ψ 2,0 (t ),...,ψ 2,M −1 (t ),...,ψ 2k −1 ,0 (t ),...,ψ 2k −1 ,M −1 (t )]T
(4)
The integration of the vector Ψ (t ) defined in Eq.(4), can be obtained as t
∫ Ψ (t ′)dt ′ = PΨ (t ) 0
(5)
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Where P is the (2 k −1 M ) × (2 k −1 M ) operational matrix for integration and is given in [24] as ⎡L ⎢ T 0 P = k ⎢ 2 ⎢M ⎢ ⎣0
L L O L
H L M 0
H ⎤ H ⎥⎥ M ⎥ ⎥ L ⎦
(6)
In Eq.(6) H and L are M × M matrices given by ⎡2 ⎢0 H = ⎢ ⎢M ⎢ ⎣0 ⎡ ⎢ 1 ⎢ ⎢− 3 ⎢ 3 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 L=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢⎣
−
0
L
0 M
L O
0
L
0⎤ 0 ⎥⎥ M⎥ ⎥ 0⎦
1 3
0
0
0
L
0
0
3 3 5
0
0
O
0
0
5 5 7
O
0
5 5 3
0 0
7 7 5 O O
0
O
0
0
O
0
0
0
L
0
0
0
−
0 O O O 0 0
0
−
2M − 1 ( 2 M − 1) 2 M − 3
⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ ⎥ 2M − 3 ⎥ ( 2 M − 3) 2 M − 1 ⎥ ⎥ ⎥ 0 ⎥⎦ 0
2.3. The Delay Operational Matrix
The delay function Ψ (t − d ) is the shift of the function Ψ ( t ) defined in Eq.(4), along the time axis by d. The general expression is given by Ψ ( t − d ) = T d Ψ (t )
(7)
Where Td is the delay operational matrix of Legendre wavelets. To find Td , we first choose q in the following manner: T ⎧ T If is integer ⎪ d ⎪ d q=⎨ ⎪ T Otherwise ⎪⎩ [ d ] + 1
Where [.] denotes greatest integer value.
(8)
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The time delay matrix Td with the dimension rs × rs where r , s are given below is given by:
⎡ 0 ( β ,α ) I ( β , β ) ⎤ Td = ⎢ ⎥ ⎣ 0 ( α ,α ) 0 ( α , β ) ⎦ ( rs × rs ) For finding α , β , first we must obtain q in Eq.(8), then: r = N ×q
N = 1,2,3,....
(9)
(10)
And with incresing N, a better approximation can be obtained:
α = N×s β = rs − α
(11)
The values of r,s of Legendre wavelets are: r = 2 k −1 , s = M
(12)
2.4. Problem Statement A linear time invarient delay system is considered as x& (t ) = Ax(t ) + Bx(t − d ) + Cu (t ) + Eu (t − d ) x (t ) = g (t ) ⎫ −d ≤t
